Groundwater pollution by organic compounds: a two-dimensional analysis of contaminant transport in stratified porous media with multiple sources of non-equilibrium partitioning

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS

Int. J. Numer. Anal. Meth. Geomech., 23, 1717}1732 (1999)

GROUNDWATER POLLUTION BY ORGANICCOMPOUNDS: A TWO-DIMENSIONAL ANALYSISOF CONTAMINANT TRANSPORT IN STRATIFIED

POROUS MEDIA WITH MULTIPLE SOURCESOF NON-EQUILIBRIUM PARTITIONING

ABBAS H. ELZEIN* AND JOHN R. BOOKER

Centre for Geotechnical Research, Department of Civil Engineering, University of Sydney, NSW 2006, Australia

SUMMARY

Transport of pollutants in soil and groundwater often occurs in strati"ed media under non-equilibriumconditions. Con"ned aquifers are usually bounded by low-permeability layers of soil which have been shownto exert a signi"cant in#uence on the fate of contaminants in groundwater. Numerical solutions of transportequations have usually been restricted to single layers and have included single sources of non-equilibriumprocesses or none at all. The e!ect of soil strati"cation itself has sometimes been reduced to a transport-based non-equilibrium process.

A boundary element solution of the transport equations in the Laplace domain is extended to includemultiple sources of non-equilibrium processes in saturated media under the assumption of rate-limited masstransfer. Green functions accurately model in"nite and semi-in"nite domains such as soils and Laplacetransforms remove the need for time-stepping and the associated numerical complexity. The proposednumerical technique is validated by comparing its results to analytical solutions. Its scope is illustratedthrough a case study of a sand aquifer bounded by less permeable layers of silt, and in"ltrated by pollutantsfrom a neighbouring lake. Copyright ( 1999 John Wiley & Sons, Ltd.

KEY WORDS: contaminant migration; non-equilibrium sorption; di!usion advection; boundary elementmethod; Green's functions; porous media

INTRODUCTION

Contaminant transport in soil and groundwater aquifers have come under close scienti"c scrutinyover the last decade. This is due to tighter environmental legislative requirements re#ectinga greater awareness of the scale of groundwater pollution, but also to a wider application ofbio-remediation techniques to contaminated sites. In particular, the leakage of gasoline fromservice stations and transmission pipes occurs frequently enough to constitute a likely andworrying cause of groundwater pollution.1~4

Experimental investigations have revealed that a Local Equilibrium Assumption (LEA),resulting in instantaneous rather than time-dependent sorption, sometimes fail to represent the

*Correspondence to: A. H. Elzein, Centre for Geotechnical Research, Department of Civil Engineering, University ofSydney, NSW 2006, Australia

CCC 0363}9061/99/141717}16$17.50 Received 22 September 1997Copyright ( 1999 John Wiley & Sons, Ltd. Revised 8 June 1998

observed behaviour adequately. The study of a number of compounds, particularly organics suchas tetrachloroethylene, toulene and xylenes, has shown that a signi"cant deviation from equilib-rium may occur, particularly under conditions of high hydraulic conductivity.5~9 Models con-taining simple rate-limited non-equilibrium component of mass transfer have succeeded inaccounting for such deviations.10~12

Non-equilibrium behaviour has been found to be either sorption related or physical i.e.transport related.13 Sorption-related non-equilibrium processes may result from rate-limitedchemical sorption at certain sorption sites or may be due to slow intra-organic or intra-mineraldi!usive mass transfer. The former a!ects site-speci"c partitioning of non-hydrophobic organicchemicals and inorganic solutes.14 Physical non-equilibrium processes may be identi"ed througha distinction between &mobile' and &immobile' regions of the soil, where advection transportoccurs in the "rst but not the second.15,16 The mobile/immobile bi-continuum appears to bea valid approximation of the non-uniformity of the advective "eld, particularly in aggregatedheterogeneous soils. Such non-uniformity may either operate at the macro-pore level, indicatinglocal variation of hydraulic conductivity or, it may be due to soil strati"cation frequentlyencountered around soil aquifers.10 The &immobile' fraction is meant to denote, in addition toimmobile or quasi-stagnant pore water, residual non-aqueous phase liquids. The exchangebetween &mobile' contaminants and &immobile' regions is believed to be governed by di!usionlaws. Thus, the immobile fraction, in its interaction with the contaminant, exhibits sorption-likebehaviour which is mathematically equivalent to the more conventional rate-limited non-equilib-rium chemical sorption. The explicit modelling of di!usion within the immobile fraction has beensuccessfully attempted by a number of authors; however, this approach does not seem to o!era signi"cant analytical advantage over mathematical rate-limited models, chie#y because of thedi$culty of estimating parameters re#ecting pore shape.9,17~19

Thus, strong experimental evidence suggests that a series of non-equilibrium processes, allamenable to rate-limited "rst-order mass transfer, play an important role in contaminanttransport in soils and must therefore be included in the modelling e!ort.

While "nite-di!erence schemes are most commonly used for solving the non-equilibriumtransport equations,8,9,14,20 "nite element and analytical solutions have also been successfullyused.11,21,22 Most of these solutions were developed in order to simulate experimental investiga-tions and are therefore one-dimensional in space. Leij et al.,23 on the other hand, developedanalytical solutions for simple problems in 3-D. Mayer and Miller24 developed a two-phasemodel of the coupled #ow/transport equations, to study the characteristics of NAPL/aqueousphase mass transfer in heterogeneous media and compared predictions based on local equilib-rium assumption to those of non-equilibrium formulations. The method is a "nite elementtechnique combined to a "nite-di!erence time-stepping scheme. Sorption onto the solid phasewas neglected and the computational e!ort required to solve four second-order coupled equa-tions could only be met by a supercomputer capability, whereby around 20 simulations con-sumed over 1000 h of CRAY-YMP time. To the best of the authors' knowledge, no generalsolutions of the transport equations with both chemical and physical non-equilibrium partition-ing have been developed in 2-D or 3-D, that are applicable to arbitrary soil strati"cations andsource shapes. The purpose of this paper is to propose such a solution in 2-D using a boundaryelement approach. The 3-D extension, which has also been carried out, will be described ina subsequent paper.

The Boundary Element Method (BEM)25,26 emerged in the late 1970s as a powerful alternativeto the more classical "nite element and "nite di!erence methods. Through the use of Green

1718 A. H. ELZEIN AND J. R. BOOKER

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech., 23, 1717}1732 (1999)

identities and known singular solutions, the method o!ers the possibility of reducing thedimensionality of the problem by one and therefore considerable savings in preprocessing e!orts.The BEM has been successfully applied to many #ow problems in porous media including thedi!usive/advective transport equation.27 However, its application to non-linear and transientproblems remained somewhat problematic because of the di$culty of developing adequatesingular solutions and the existence of domain integrals in the formulations. These obstacles havebeen addressed in di!erent ways by di!erent authors but the resulting solutions often sacri"cedthe relative simplicity and elegance of linear steady-state solutions. The Laplace transform hasbeen applied to di!erential equations of time-dependent thermoelasticity28~30 in order toeliminate the time variable and successfully develop a computationally e$cient boundaryelement solution. Leo and Booker31,32 combined a semi-analytical Laplace solution of thecontaminant migration problem33,34 to a boundary element solution of the modi"ed Helmholtzequation and developed an e$cient technique for solving the 2-D contaminant transportproblems with anisotropic di!usion properties in heterogeneous media. The formulation appliesthe Laplace transform to the governing equations then eliminates the di!usion anisotropy andadvection terms through a co-ordinate transform combined to an exponential representation ofthe state variable. A standard complex-valued modi"ed Helmholtz equation is thus obtained,which is solved by BEM and inverted back into the time-domain numerically. The method onlyrequires boundary discretization and does not necessitate time stepping. Its e$ciency is furtheremphasised in multi-layer domains as the computational cost of adding extra layers is relativelylow.

The present paper extends this boundary element solution to cover multiple sources ofnon-equilibrium processes in saturated heterogeneous media. The proposed technique isvalidated by comparing its predictions to analytical solutions developed elsewhere. In addition,the scope of the method is illustrated through the study of contaminant migration in a sandaquifer, bounded by less permeable silt layers, and in"ltrated by pollutants from a neighbouringlake.

BOUNDARY ELEMENT SOLUTION OF TRANSPORT EQUATIONSIN SATURATED MEDIA WITH NON-EQUILIBRIUM PARTITIONING

Concentration of contaminant dissolved in the aqueous component of a saturated two-dimen-sional porous domain ) with boundary !, is assumed to be governed by the following processes:Fickian-type di!usion representing both molecular di!usion and mechanical dispersion, advec-tive steady-state laminar #ow, instantaneous and rate-limited partitioning between dissolved andsolid phases, and rate-limited exchange with an immobile fraction of the soil's solution. A sche-matic illustration of the non-equilibrium model adopted here is given in Figure 1. The processesshown are assumed to take place at every point of the soil and re#ect the division of thecontaminant among the three phases: dissolved in mobile water (block A), dissolved in immobilewater or non-aqueous liquid (block B) or sorbed onto the soil's mineral or organic matrix (blockC and D). A fraction F of sorption occurs instantaneously, while the remaining fraction isgoverned by a "rst-order mass-transfer equation. The &sorbed' contaminant block C and D areclearly in parallel, indicating that &equilibrium' and &non-equilibrium' processes are simultaneousand independent. The &immobile solution' block B is assumed to undergo no partitioning with thesolid phase. Hence, the &immobile' block acts as an additional non-equilibrium sorption site. Aspointed out by Brusseau et al.14, although multi-source models are usually mathematically

GROUNDWATER POLLUTION BY ORGANIC COMPOUNDS 1719


Figure 1. Model for non-equilibrium processes

reducible to a single non-equilibrium component, the explicit representation of multiple sourcesof rate-limited exchange, allows for more realistic transport and sorption parameters to be used inforward analyses, and to be calculated in best-"t inverse solutions. Strictly, the &mobile' fraction isdistinguished from the &immobile' one by the existence of a signi"cant advection transportcomponent in the "rst but not the second.

Applying mass-conservation principles and a Fickian di!usion relationship, the standarddi!usion/advection equation can be derived. As indicated earlier, exchanges between the solidphase and the mobile solution are divided into equilibrium and non-equilibrium components.The selected co-ordinate system is assumed to be aligned with the direction of groundwater #ow.Biodegradation and radioactive decay are neglected. The following coupled equations areobtained:

n.D

xx

L2c.

Lx2#n

.D

yy

L2c.

Ly2!n

.<!x

Lc.

Lx!n

.<!y

Lc.

Ly"(n

.#o

$FK

$)Lc

.Lt

#o$

Lq.

Lt#n

*.

Lc*.

Lt

(1)

Lq.

Lt"K

tMK

$(1!F )c

.!q

.N (2)

n*.

Lc*.

Lt"w (c

.!c

*.) (3)

(at steady-state: q."K

$(1!F )c

., c

*."c

.)

where t(T) is time, x (L) and y(L) denote a 2-D co-ordinate system, c.(t, x, y) (M/L3) is the

dissolved concentration of contaminant in mobile water, c*.

(t, x, y) (M/L3) is the dissolvedconcentration of contaminant in immobile water, D

xx(L2/T) and D

yy(L2/T) are the coe$cients of

hydrodynamic dispersion of contaminant in soil in the x and y directions, respectively, <!x

(L/T)and <

!y(L/T) are the values of the groundwater velocity in the x and y directions, respectively,

n.

is the mobile-water-"lled porosity of the soil, o$(M/L3) is the density of the dry soil, n

*.is the



immobile-water-"lled porosity of the soil, q.

(M/M) is the non-instantaneous fraction of sorbedmaterial, K

$(L3/M) is the soil/contaminant partitioning constant, F is the fraction of steady-

state sorption that is instantaneous, Kt(1 /T) is the "rst-order sorption rate constant, and w (1/T)

is the "rst-order mass transfer coe$cient between mobile and immobile water.The sorption retardation factor is de"ned as: R"1#o

$K

$/n

.. Equations (2) and (3) are

"rst-order, rate-limited partitioning relationships where steady-state equilibrium is approachedat a given rate, K

tor w.

A general mathematical equation representing a number of widely occurring boundary condi-tions can be written as follows:

ac.#b f

.n"u (4)

where c.

is the pollutant concentration in mobile water at the boundary, f.n

is the #ux ofcontaminant dissolved in mobile-water in the direction n of the outward normal to the boundary,and a, b and u are given coe$cients which may vary with position. Equation (4) can clearlyaccommodate both a speci"ed concentration, a speci"ed #ux or any linear combination of both.Furthermore, the normal #ux can be expressed as a function of c

.and its gradient in the normal

direction Lc./Ln using Fick's law; therefore equation (4) can also represent gradient boundary

conditions. Another boundary condition of interest is the waste-repository or ,nite-mass equa-tion33,34 which conserves a speci"ed mass of contaminant rather than maintaining a speci"ed#ux or concentration. Such a constraint is often a more accurate representation of contaminantinput into soil, as the amount of spill or leakage is usually "nite over the time scales of interest.Assuming that the contaminant source is well mixed and no retardation, biodegradation orradioactive decay occur within it, and that the concentration in the repository is an unspeci,ed(i.e. unknown) quantity that is spatially uniform, the "nite-mass condition can be expressed asfollows:

<r(cr

.!cr

.0)#P

t

0P!r

f r.n

d!rdt"0 (5a)

where !ris the boundary at which the condition applies, <

ris the volume of contaminant mass,

cr.(t) is the concentration at !

rassumed uniform, cr

.0is the concentration at t"0 and f r

./is the

#ux of contaminant through !r. Dividing equation (5a) by <

r, the boundary condition can be

expressed in terms of the equivalent height Hf

de"ned as the height of contaminant source perunit plan area. If ¸

ris the length of the repository's boundary then (5a) becomes

cr.!cr

.0#

1

Hf¸

rP

t

0P!r

f r.n

d!rdt"0 (5b)

Introducing the following Laplace transforms:

(cN., cN

*., qN

.fM.x

, fM.y

)"P=

0

e~st(c., c

*., q

., f

.x, f

.y) dt (6)

using Laplace transform properties and rearranging, equations (1)}(5) become

Dxx

L2cN.

Lx2#D

yy

L2cN.

Ly2!<

!x

LcN.

Lx!<

!y

LcN.

Ly"'cN

.!'

.c.0

!

n*.

n.

'*.

c*.0

(7)



where

'"A'.#

o$

n.

't#

n*.

n.

'*.B s

'."1#

o$FK

$n.

'*."

w

n*.

s#w

't"

KtK

$(1!F )

s#Kt

qN."'

tcN.

(8)

cN*."'

*.AcN.#

n*.w

c*.0B (9)

acN.#b fM

.n"

u

s(10)

cN r."

cr.0s

!

1

Hf¸

r

1

s P!rfM r.n

d!r

(11)

(c.0

and c*.0

are the initial concentrations in the mobile and immobile solutions respectively, andare taken as equal to satisfy initial equilibrium).

Equation (7) can be transformed into a modi"ed Helmholtz equation by applying a dualtransform.35 The "rst transform is a change of co-ordinate system which removes the anisotropyof the di!usion and dispersion properties. The second transform introduces an exponentialrepresentation of cN

.which removes the advective terms from the di!erential equation. Assuming

hydrodynamic dispersion and advection coe$cients are uniform within a soil layer, the followingequation is obtained:

Da+2cN

.a"(cN

.a(12)

A+2"L2

LX2#

L2

L>2Bwhere, for the "rst transform:

X"uxx, >"u

yy, u

x"J(D

xx/D

a), u

y"J(D

yy/D

a), D

!"J(D

xx/D

yy), <

X"u

y<x,

<Y"u

x<y, <

N"<

X¸

X#<

Y¸

YfM.N

"<NcN.!D

!

LcN.

LN, fM

.N"(u

ylx¸

X#u

xly¸Y) fM

.n

(13)



and for the second transform:

aX"

<X

2D!

aY"

<Y

2D!

cN."cN

.ae(aXX`aYY)

fM.N

"fM.aN

e(aXX`aYY)

("'#D!(a2

X#a2

Y)

fM.aN

"

<N

2cN.a

!D!

LcN.a

LN(14)

lxand l

yare the direction cosines of the normal n to the boundary in the (x, y) coordinate system,

N is the normal direction to the boundary in the (X, > ) coordinate system, ¸X

and ¸Y

are thedirection cosines of the normal N. Identically zero initial distribution of contaminants has beenassumed in equation (12). Simple con"gurations of initial concentrations such as uniform onescan be easily dealt with by deriving the corresponding particular solutions and writing equation(12) in terms of incremental concentrations. For more elaborate or random distributions, othertechniques, such as the dual-reciprocity method,36 have been successfully used to avoid domainintegration in both potential and elasticity problems. In this paper, only problems with zeroinitial concentrations will be considered.

Boundary element solutions of equation (12) are widely documented in the literature.25~27

Using the Gauss divergence theorem and singular solutions of equation (12):

e(r0)cN

.a0"P!

(cN.a

fM *.aN

!cN *.a

fM.aN

) d! (15)

where r0

is the position vector of the singularity source, cN.a0

is the value of cN.a

at the singularitysource, and e (r

0) is 1 when the source is inside the domain and 1/2 on a smooth boundary.

A fundamental solution McN *.a

(r), f *.aN

(r)N of the two-dimensional modi"ed Helmholtz equation inthe Laplace domain can be written as follows:

cN *.a

(r)"1

2nD!

[K0(rJ((/D

!))] (16)

fM *.aN

(r)"<N2

cN *.a

!D!

LcN *.a

LN(17)

where r is the Euclidean distance between the source node and the integration point and K0(R) is

the modi"ed Bessel function of a given variable R.Functions of concentration and normal #uxes can be discretised by dividing the boundary of

the problem into n%straight- or curvilinear-line elements j over which system variables cN

.aand

fM.aN

are represented as functions of their values cN i.a

and fM i.aN

at n1pre-selected nodes i, by means of



simple interpolation functions uji, where for every element j

(cN.a

, fM.aN

)"np+i/1

uji(cN i

.a, fM i

.aN) ( j"1, n

%) (18)

In the present implementation, a library of constant (n1"1), linear (n

1"2) and quadratic

(n1"3) elements is developed. The problem of multiply de"ned normals to the boundary at

corners has received particular attention in boundary element literature and di!erent ways ofdealing with discontinuities of both geometrical (corners) and physical (e.g. discontinuity ofsource or sink) nature have been proposed. Partly discontinuous elements are used in the presentformulation to eliminate multiple-de"nitions of fM

./. Typically, the node on the element-edge

occurring at a corner is located inside the element at some small, arbitrary distance from the edge.Interpolation functions for such elements are derived numerically rather than semi-analytically.

If N5total nodes are created over the entire boundary of the problem, then N

5equations can be

constructed by placing the singularity source point at each of these nodes and writing equation(15) accordingly. A gaussian numerical integration scheme is used to evaluate the integrals.However, singularities log r and 1/r are isolated and integrated analytically. Replacing variablesc6.a

and fM.aN

with expressions in terms of cN.

and fM.n

, respectively (equations (13) and (14)), andrearranging, the following set of equations is obtained:

[H]McN k.N"[G]M fM k

.nN (k"1, N

5) (19)

By applying boundary conditions of the form (10) or (11), system (19) can be reduced to a set ofN

5algebraic equations with N

5unknowns that can be solved by a standard Gaussian solution

procedure.37 The values of c.

at each node can be obtained by inverting the solution intotime-domain numerically. An accurate algorithm proposed by Talbot,38 using complex values ofthe Laplace variable s, is used for this purpose.

Di!erent layers of soils with di!erent transport and partitioning properties can be de"ned asseparate zones and a separate set of equations and unknowns is developed for each zone. At theinterface between two zones, forcing the continuity of the cN

.and fM

.nunknowns provides

additional equations in lieu of the boundary conditions applied at non-interface boundaries.Thus, sets of zone-matrices contribute to the construction of global matrices re#ecting the entireproblem. Zoning is also an e$cient way of creating a computationally e$cient bandwidth inotherwise fully-populated boundary element matrices. The extra cost of adding a zone to theproblem, already relatively low because it involves only additional boundary discretization, isfurther reduced by the implicit creation of a bandwidth. A medium made of layers of soil istherefore an especially suitable problem for analysis by this technique.

The algorithm has been implemented in FORTRAN on UNIX and PC platforms (programCONTAN), along with a text-based easy-to-use facility for data-input and a number of output"les for viewing on a spreadsheet. Meshing is performed automatically subject to user speci"ca-tions. A convergence analysis has been performed on a number of simple problems for whichanalytical solutions have also been derived. Both the number of elements and their order ofaccuracy, from constant to quadratic, have been increased in order to verify the convergence andstability of the solution. Analytical and BEM results for two problems are compared in the nextsection: a leaking cylindrical repository burried in an in"nite homogeneous soil, and a strati"edsoil subjected to a "nite mass of contaminant at its surface. An aquifer bounded by less permeablestrata and polluted by a neighbouring lake is also analysed with CONTAN to illustrate the scope



of the method. All analyses were ran on a standard 90 MHz PC with 32 Mbytes of memory.Analysis time varied between a few seconds and under 3 h.

RESULTS

Deeply buried cylindrical source

A leaking cylindrical repository of radius r0"1 m, deeply buried in a homogeneous soil and

containing a "nite mass of contaminant, is analysed with CONTAN using 40 constant elements.A section-view of the problem is shown in Figure 2(a). An analytical solution for homogeneoussoils with isotropic di!usion properties, proposed by Rahman and Booker,39 is extended here toinclude non-equilibrium partitioning. The following parameters are used: c

.r0"3000 mg/l,

Dx"D

y"0)01 m2/y, v

x"v

y"0, n

."0)4, n

*."0, R"5, K

t"1/yr, F"0)4. This is therefore

a case of non-equilibrium sorption with no immobile-water fraction. c.

is assumed to vanish atin"nity. Figure 2(b) shows the variation of c

.along a vertical line in the soil, 3 months after the

beginning of leakage. The same curve, obtained for the case of instantaneous equilibrium, is alsoincluded to demonstrate the e!ect of non-equilibrium sorption. The "gure compares boundaryelement predictions to analytical results. There is clearly good agreement between the twosolutions.

Stratixed soil

Next, the case of a heterogeneous, multi-layer soil, is considered. A section view of the soil isshown in Figure 3(a). Two layers of sand and silt are underlain by an aquifer, with a largehorizontal #ow rate. The soil is subjected to a "nite-mass of contaminant at its surface anda relatively low value of K

tcorresponding to slow sorption. Contaminant reaching the aquifer is

assumed to be instantly washed away, and concentration c.

at the bottom of the sand layer isthus taken to be zero. The top layer is described by di!usion coe$cients D

y"5 m2/yr and

Dx"0, retardation coe$cient R"2 and porosity n

."0)35. The parameters of the bottom layer

are: Dy"0)5 m2/yr and D

x"0, retardation coe$cient R"4 and porosity n

."0)4. This is

a one-dimensional problem which can be solved analytically by extending a technique suggestedby Rowe and Booker.33 Both instantaneous and non-equilibrium sorption using K

t"1/yr

Figure 2(a). Section view of buried cylindrical repository



Figure 2(b). Buried cylindrical repository (centre at the origin of axes): c.

along the y-axis, 3 months

Figure 3(a). Multi-layer soil

and F"0)3 are considered. The two layers of soil are represented with two separate zones havingdi!erent physical properties. A total of 30 constant elements along the three horizontal boundarylines of 10 m width each and 16 quadratic elements over vertical boundaries, are used. Thus allelements are 1 m long. Figure 3(b) illustrates the e!ect of non-equilibrium sorption on thec.

pro"les at di!erent times, while Figures 3(c) and 3(d) compare BEM predictions to analyticalresults for c

.pro"les and breakthrough-curves, respectively. As expected, the e!ect of non-

equilibrium sorption diminishes with time and by year 5 has almost disappeared. Predictionsneglecting non-equilibrium processes are clearly non-conservative. Excellent agreement betweenthe two methods of solution is again obtained. The di!erence in di!usion properties between thetwo layers of soil translates into a slope discontinuity in the pro"le curves at the interface.

Sand aquifer bounded by silt layers and polluted by a neighbouring lake

Finally, to illustrate the scope of the method, the case of an aquifer, bounded by two layers ofless permeable silty soil, has been considered. The aquifer is polluted by a neighbouring lakemaintained at a constant concentration c

0of contaminant in mobile water (Figure 4(a)). The



Figure 3(b). E!ect of non-equilibrium sorption on c.

along depth

Figure 3(c). Accuracy of BEM: c.

along depth at di!erent times

problem is analysed with CONTAN using a total of 172 quadratic elements of around 1 m lengtheach over a total width of 40 m and total depth of 16 m (Figure 4(b)). Parameters used in theanalysis are shown in Table I. Non-equilibrium rate-constants K

tand w reported in the literature

vary widely. Performing "eld experiments on organic solutes, Goltz and Roberts6 used transfer-rates from mobile to immobile regions, close to 1/yr. In laboratory experiments aimed at



Figure 3(d). Accuracy of BEM: breakthrough curves at the interface between the two layers (y"4 m)

Figure 4(a). Aquifer bounded by silt strata

analysing benzene and toluene transport in soils containing residual hydrocarbon, under condi-tions of very high hydraulic conductivity, Hat"eld et al.12 calculated best-"t rates in the order of2000/yr to 1 00 000/yr. Relatively slow mass-transfer has been simulated in this example(K

t"w"1/yr) in order to better appreciate the potential deviation from equilibrium. To

illustrate the combined e!ect of the di!usion, advection and sorption capacities of the silt layers,a second analysis is performed for a fully con"ned aquifer, i.e. assuming that there is nomass-transfer from or into the silt strata. In addition, the e!ects of the mobile fraction in theaquifer is demonstrated by including the results from a third analysis performed with n

*."0 and

n."0)36, assuming that is, all the soil porosity is occupied by mobile water. Finally, a fourth

analysis considers conditions of instantaneous sorption in the silt layers thereby isolating thee!ects of non-equilibrium sorption. Figure 4(c) depicts the contaminant concentration in theaquifer along the line y"2 m at di!erent times and Figure 4(d) shows the breakthrough curves at14 m from the pollution source up to 5 years.



Figure 4(b). Boundary element mesh

Figure 4(c). c.

along aquifer-length at various times

Predictably, the existence of an immobile fraction reduces the concentration in mobile water asthe mobile-water porosity is reduced and immobile-water regions act as additional sorption sites.Non-equilibrium sorption in the silt layers has no apparent e!ect on c

.because relatively slow

transport processes allow enough time for equilibrium to be reached. Finally, the combined e!ectof di!usion, advection and sorption in the silt layers can be seen in the 2-yr curves and moreclearly in the 5-yr curves of Figure 4(c). After 5 years, 14 m away from the pollution source, c

.of

the multi-strata analysis is about 11% lower than its amount calculated for the case of a fullycon"ned aquifer (Figure 4(d)).

CONCLUSIONS

An e$cient boundary element method is proposed for the solution of the 2-D contaminanttransport equations, containing multiple source of non-equilibrium processes. The method candeal with heterogeneous media and anisotropic transport and partitioning parameters, and has



Figure 4(d). breakthrough curves at x"14 m, y"2 m

Table I. Parameters for sand-acquifer problem

Top layer Aquifer Bottom layer

Dx

(m2/yr) 0)1 5 0)1D

y(m2/yr) 0)01 0)5 0)01

vx

(m/yr) 0)1 5 0)1vy(m/yr) 0)0 0)0 0)0

n.

0)4 0)26 0)4n*.

0)0 0)1 0)0R 4 1)0 4F 0)2 na 0)2K

d(m3/kg) 0)0006977 0)0 0)0006977

Kt(/yr) 1 na 1

w (/yr) na 1 na

na: Not applicable.

small computing requirements. The method has been validated by comparing its predictions toanalytical results for single-layer and multi-layer soils. A simultaneous inclusion of time-depen-dent sorption and residual contaminated immobile fraction, allows engineers to estimate moreaccurately the pro"le of contaminants in soil, particularly organic compounds.

REFERENCES

1. OECD, &Biotechnology for a clean environment. Prevention, detection, remediation', Organisation for EconomicCo-Operation and Development, (1994).



2. J. R. Matis, &Petroleum contamination of groundwater in Maryland', Ground=ater, 9(6), 57}61 (1971).3. J. O. Osgood, &Hydrocarbon dispersion in groundwater: signi"cance and characteristics', Ground =ater, 12 (6),

427}438 (1974).4. V. Pye and J. Kelly, &The extent of groundwater contamination in the United States', Groundwater Contamination,

National Academy Press, (1984) pp. 23}44.5. J. R. Pickens, K. J. I. Jackson and W. F. Meritt, &Measurement of distribution coe$cients using a radial injection

dual-tracer test',=ater Resour. Res., 17 (3), 529}544 (1981).6. M. N. Goltz and P. V. Roberts, &Interpreting organic solute transport data from a "eld experiment using physical

nonequilibrium models', J. Contaminant Hydrol., 1, 77}93 (1986a).7. R. C. Borden and M. D. Piwoni, &Hydrocarbon dissolution and transport: a comparison of equilibrium and kinetic

models', J. Contaminant Hydro., 10, 309}323 (1992).8. R. S. Kookana, R. Naidu and K. G. Tiller, &Sorption non-equilibrium during Cadmium transport through soils', Aus.

J. Soil Res., 32, 635}651 (1994).9. L. A. Gaston and M. A. Locke, &Organic chemicals in the environment. Fluometuron sorption and transport in

Dundee soil', J. Environ. Qual., 24, 29}36 (1995).10. M. L. Brusseau, &Application of a multi-process nonequilibrium sorption model to solute transport in a strati"ed

porous medium',=ater Resour. Res., 27(4), 589}595 (1991).11. D. W. Smith, R. K. Rowe and J. R. Booker, &The analysis of pollutant migration through soil with linear hereditary

time-dependent sorption', Int. J. Numer. Anal. Meth. Geomech., 17, 225}274 (1993).12. K. Hat"eld, J. Ziegler and D. R. Burris, &Transport in porous media containing residual hydrocarbon. II: Experiments,

J. Environ. Engng.l, 119(3), 559}575 (1993).13. M. L. Brusseau and P. S. C. Rao, &Sorption nonideality during organic contaminant transport in porous media', CRC

Crit. Rev. Environ. Control, 19 (1), 33}99 (1989).14. M. L. Brusseau, R. E. Jessup and P. S. C. Rao, &Modelling the transport of solutes in#uenced by multiprocess

non-equilibrium',=ater Resour. Res., 25, 1971}1988 (1989).15. K. H. Coats and B. D. Smith, &Dead-end pore volume and dispersion in porous media', Soc. Pet. Engrs. J., 4, 73}84

(1964).16. M. Th. Van Genuchten, &A general approach for modeling solute transport in structured soils', Proc. Hydrogeology of

¸ocks with ¸ow Hydraulic Conductivity, Memories of the International Association of Hydrogeologists, Vol. 17(part 1), pp. 513}526.

17. P. S. C. Rao, R. E. Jessup, D. E. Rolston, J. M. Davidson and D. P. Kilcrease, &Experimental and mathe-matical description of nonadsorbed solute transfer by di!usion in spherical aggregates', J. Soil Sci. Soc., 44, 684}692(1980).

18. J. C. Parker and A. J. Valocchi, &Constraints on the validity of equilibrium and "rst-order kinetic transport models instructured soils',=ater Resour. Res., 22, 399}407 (1986).

19. T. C. Harmon, L. Semprini and P. V. Roberts, &Simulating solute transport using laboratory-base sorption para-meters', J. Environ. Engng., 118(5), 666}689 (1992).

20. P. Nkeddi-Kizza, M. L. Brusseau, P. S. C. Rao and A. G. Hornsby, &Nonequilibrium sorption during displacement ofhydrophobic organic chemicals and 45Ca through soil columns with aqueous and mixed solvents', Environ. Sci.¹echnol., 23, 814}820 (1989).

21. M. Th. Van Genuchten and P. J. Wierenga, &Mass transfer studies in sorbing porous media: analytical solutions',J. Soil Sci. Soc., 40, 673}679 (1976).

22. M. N. Goltz and P. V. Roberts, &Three-dimensional solutions for solute transport in in"nite medium with mobile andimmobile zones',=ater Resour. Res., 22, 1139}1148 (1986b).

23. F. J. Leij, N. Toride and M. Th. van Genuchten, &Analytical solutions for non-equilibrium solute transport inthree-dimensional porous media', J. Hydrol., Vol. 151, 193}228 (1993).

24. A. S. Mayer and C. T. Miller, &The in#uence of mass transfer characteristics and porous media heterogeneity onnonaqueous phase dissolution',=ater Resour. Res., 32(6), 1551}1567 (1996).

25. P. K. Banerjee and R. Butter"eld, Boundary Element Methods in Engineering Science, McGraw-Hill, London, 1981452 p.

26. C. A. Brebbia, ¹he Boundary Element Method for Engineers, Pentech Press, London, 1978 189 p.27. J. A. Liggett and P. L.-F. Liu, ¹he Boundary Integral Equation Method for Porous Media Flow, Georges Allen

& Unwin, London, (1983) 255 p.28. T. A. Cruze and F. J. Rizzo, &A direct formulation and numerical solution in the general transient elasto-dynamic

problems I', J. Math. Anal. Appl., 22, 244}259 (1968).29. D. W. Smith and J. R. Booker, &Boundary element analysis of transient thermoelasticity', Int. J. Numer. Anal. Meth.

Geomech., 13, 283}302 (1988).30. D. W. Smith, &Boundary integral methods for thermoelastic problems', Ph.D. ¹hesis, University of Sydney, (1990)

292 p.31. C. J. Leo and J. R. Booker, &A boundary element method for the analysis of contaminant transport in porous media I:

homogeneous porous media', Int. J. Numer. Anal. Meth. Geomech., 23, 1681}1699 (1999).



32. C. J. Leo and J. R. Booker, &A boundary element method for the analysis of contaminant transport in porous media II:heterogeneous porous media', Int. J. Numer. Anal. Meth. Geomech., 23, 1701}1715 (1999).

33. R. K. Rowe and J. R. Booker, &1-D pollutant migration in soils of "nite depth', J. Geotech. Engng., 111(4), 479}499(1985a).

34. R. K. Rowe and J. R. Booker, &Two-dimensional migration in soils of "nite depth', Can. Geotech. J. 22, 429}436(1985b).

35. C. J. Leo, &Boundary element analysis of contaminant transport in porous media, Ph.D. ¹hesis, University of Sydney,(1994) 353 p.

36. C. A. Brebbia and D. Nardini, &Dynamic analysis in solid mechanics by an alternative boundary element procedure',Int. J. Soil Dyn. Earthquake Engng., 2(4), 228}233 (1983).

37. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical Recipes in FOR¹RAN. ¹he Art ofScienti,c Computing, 2nd edn., Cambridge University Press, Cambridge, 1992.

38. A. Talbot, &The accurate numerical integration of Laplace transforms', J. Inst. Math. Appl., 23, 97}120 (1979).39. M. S. Rahman and J. R. Booker, &Pollutant migration from deeply buried repositories', Int. J. Numer. Anal. Meth.

Geomech., 13, 37}51 (1989).40. K. Hat"eld and T. B. Stau!er, &Transport in porous media containing residual hydrocarbon. I: Model', J. Environ.

Engng., 119(13), 540}558 (1993).41. M. T. Van Genuchten, &Non-equilibrium transport parameters from miscible displacement experiments', Research

Report No. 119, U.S. Dept. of Agriculture Science and Educational Administration, U.S. Salinity Laboratory,Riverside, California, (1981).



Documents

Groundwater pollution by organic compounds: a two-dimensional analysis of contaminant transport in stratified porous media with multiple sources of non-equilibrium partitioning